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Understanding ordered-pair solutions is fundamental in solving a system of linear equations. An ordered pair, typically denoted as \( (x, y) \), is a solution to a linear system if both equations are satisfied simultaneously when these values are substituted for \(x\) and \(y\). To visualize, consider the graph with two intersecting lines; the point where they cross represents their common solution, the ordered pair. While it's tempting to think all linear systems have many solutions, this isn't always true.

An ordered-pair that solves one linear equation might not necessarily solve another, hence it's significant to test the pair against all equations in the system. This aspect marks the importance of understanding not just singular solutions but also how systems might differ in their solution sets – there isn't a one-size-fits-all answer.

Linear systems can be categorized based on the number of solutions they possess. The three types to be aware of are:

• Consistent and Independent: Here, exactly one ordered pair solves the system, represented graphically as two lines intersecting at a singular point.
• Consistent and Dependent: These systems have infinitely many solutions because the equations represent the same line. Instead of an intersection, they align perfectly, overlapping each other.
• Inconsistent: Such systems have no solutions at all, portrayed by parallel lines that never meet. This highlights that the initial assumption that every linear system has infinitely many solutions is a misconception.

Understanding these types is crucial in analyzing the number of solutions that a system may have and interpreting why the belief mentioned in the exercise could lead to misunderstanding.

A system of linear equations comprises two or more equations with two or more variables that hold a linear relationship, striving for a common solution. When plotting these equations on a graph, they take the shape of straight lines, thus the name 'linear'. The key to solving such systems lies in finding where these lines intersect – which corresponds to the solutions of the system.

To navigate through potential solutions, various methods like graphing, substitution, elimination, or matrix operations can be employed. For example, if two lines intersect at one point, that point's coordinates provide the unique solution to the system. In contrast, if the lines are identical, the system doesn't limit solutions to a single point, resulting in infinite solutions. Conversely, if we find parallel lines that never meet, we're faced with an inconsistent system devoid of solutions.